Your options will be the most open if you have your own unlocked phone that is not tied to any carrier in your home country.

Own an unlocked phone:
Ok. You have many options here and some are better than others.1. Olleh Prepaid
If you go to the KT wireless (Olleh) store and ask them for a prepaid phone (선불). It will be the best option for you if you don't talk on the phone very much.
You pay a one-time SIM card (SIM-chip) fee of 8800 KRW.
Then, you can pay 30,000KRW for 100 minutes of talk time valid for 180 days. This includes free talk time between other Olleh customers.
Each month, you can pay 14,300 KRW for 1GB of 4G data, other prices for more or less data or if you opt for 3G data. You also get free use of the Olleh wifi, which means YAY WIFI ON THE SUBWAY!

Your best option for getting this (if you have limited Korean skills) is going to an area with either a lot of foreigners or with a lot of foreign students.

We went to the Olleh store in Sinchon, but had some difficulties because the worker was a little ... challenged and not super good at service (even in Korean).
You'll need to be in Korea at least 3 days and, often, need to have your alien registration card beforehand. You might want to bring a Korean friend with you, but just go in with confidence and YOU CAN DO IT!
I ended up using my ARC to help my friend set up her account...the guy was having difficulties.

There's also a "GLOBAL" Olleh store in Gangnam and other areas throughout the country. There, they should have someone who speaks English, Chinese, Japanese, and Russian, I believe.

2. SKT (T-World) or LG U+ Prepaid
Both of these also have prepaid options.
If I were you, I would stick with either Olleh or SKT. The wireless service is better.
However, LG U+ is one of the cheapest service plans.

3. Phone contract
Getting a contract is a little more complicated. So, I guess I would recommend this only if you either call a lot or are going to be here a long time.

With your own phone, you have the ability to shop around and can use that to your advantage to try and get the best "service" (freebies you're given at the store) and the best price.

The best deal is to sign a 2-year contract (even if you are planning on leaving earlier, you can break contract without paying a fee since you own your phone). I recommend getting a longer contract because you get a monthly fee discount, the longer your service contract is.

One thing to note is sometimes if you bring your own phone to another carrier, you may be blocked from accessing the provider's WIFI service.

If you really cannot speak Korean, there are some options.
I know there is a phone store in the Yonsei Global Center, near the dormitories and KLI. There, the workers speak English, Chinese, and Japanese just due to the fact that the majority of students using the services there attend KLI.

You can also go to the Olleh Global branches. Check the Olleh site for the most updated options.

4. Shorter stays
If you're staying a shorter amount of time, there are some other SIM card options.
My friends have tried the Evergreen SIM card service and have had varying amounts of success. This is a data-only service. Their issues were not a factor of the Korean-side service, but actually a result of their American phone's service.
My friends with Verizon were able to use the 4G data, but were not able to access the included WIFI.
You can pick up your cards in Incheon Airport or Insadong.

Still others rent phones at the airport (you can get a better rate if you book ahead) or WIFI eggs. I think WIFI eggs are a very Korean thing. They're mobile hotspots that you just put in your pocket. If you have a smartphone, I personally thing the WIFI egg or the SIM card is a better option. The rental phones are often still flip phones which can get frustrating since you have to text a lot (calling costs a lot of money).

One thing to keep in mind with all these options is that accepting calls is free, regardless of the carrier the caller uses.

Do not have own unlocked phone or you want to buy a new one:

From personal experience, don't trust Verizon Wireless. I bought an unlocked, global phone from them and confirmed multiple times before coming that my Samsung Galaxy S5 would work in Korea on a Korean carrier. I got here, and after hours spent in multiple phone stores and going to the Samsung A/S center to get it checked out, was told that my American carrier locked the phone. I paid for this phone outright, but Verizon had locked it to be used only on their service.

Ok, so that being said. I have experience finding good deals on phones in Korea.

1. Noksapyeong/Itaewon
Some of my white friends really like going here. I don't really feel confident buying used phones from these shop owners since I have heard bad stories about how most of the phones are stolen, but these are options. Just be smart!

Used/New phones available

2. Yongsan/TechnoMart
You can also go here, but again, be smart about your purchases.

Use/New phones available

3. Internet
There are plenty of options
1. Craigslist - Obviously with caution
2. Various buy & sell groups on Facebook (I actually got a really good deal on an iPhone through this method, obviously you need to be cautious)
3. Naver Cafes (this is a good option if you can read Korean) - 중고나라4. Suwon Station
I don't know why, but this is one of the best places to go for phone shopping, if you're on a budget. It's kind of a trek from Seoul, but I have had good luck here getting a really good price on a brand new phone (aka it was free) with a short contract (12 months, and even after cancelling the contract I didn't pay a device fee).

You'll need an ARC and Korean bank account.

5. Gangnam Station
Not quite as cheap as at Suwon, but much closer if you're located in Seoul.
There are plenty of cellphone dealers around Gangnam station shopping underground.
There are some confusing schemes, so make sure your Korean is up to par.
Some newer schemes are kind of "leasing" a phone. You pay a ton for the first few months of your contract (then you change it to a cheaper plan), but you don't pay the phone device fee. My friend was explaining it to me...she kept using the word for "lease" but made it seem like she actually owned the phone at the end of the one-year contract.
You'll probably want to bring a Korean friend.

You'll need an ARC and a Korean bank account.6. At a mobile service provider
Ok, from an American perspective, new phones in Korea are A TON OF MONEY!
When you go to a phone store, it's quite complicated. I'll try and explain it simply here. Let me know if you have questions.

Let's pretend you want the newest iPhone or Samsung model. You'll likely pay about 35,000KRW per month on a 3-year contract for the phone itself (so, roughly 1million KRW for the phone). The company will add this amount to your monthly contract.
But, by signing a 3-year contract you get x% off of the quoted/posted monthly fee.
Usually there is some promotion where, if you sign up for the high price service contract (like 90~110,000KRW per month), you get some other discount off the phone price per month. Sometimes it's free, but not always.
However, you are not locked into that price for a full 3 years (or 2 years). If you read your contract, you'll usually be required to pay that high price for something like 6 months (just read your contract), but then you still can maintain your discounted phone price throughout the course of the rest of your contract.
I hope that makes sense!

You'll probably really want a Korean friend to go with you if you choose option 5. Even if you have good Korean skills, contracts are confusing even for Koreans.

As with what I mentioned above, if your Korean is kind of basic and you don't want to bother your friends, you can always stop at that phone store in Yonsei's KLI. It's an Olleh branch and I think they'll explain the different contracts to you and the best price (not from experience...).

Hello, I apologize for my late reply. I just changed jobs and didn't check my blog.Yes, it will be nearly impossible to get a SIM card on the weekend as a foreigner. They do not activate new SIM cards after 5 or 6PM on Fridays.